Introduction

The theory of generating function was attributed to Euler [1,2] but when used in sequence algebra, its defintion has been modified. In Table 1 both forms are shown side by side for comparisons. The similarities and differences between these two forms and areas of applications should be noted.

Table 1 - Differences Between Euler's Form and Sequence Algebraic Form ========================================================
Euler's Form.......................................Sequence Algebraic Form
.............................................................There are two forms:
Power series form:.................................Power series form:
F(z) = ao+a1x^1+a2x^2+..+anx^n+......F(z)=1.x^ao+1.x^a1+1.x^a2+..+1.xa^n+..
.............................................................Laurent series form:
.............................................................F(z) = 1/x^ao+1/x^a1+1/x^a2+..+1/x^an+..

Set Notations:......................................Set Notations:
{ao, a1, a2, ,,,,, an, ,,,,,,,}......................{1,1,1,,,,,,,,1,,,,,,,,} for both versions
The values of the elements are................The values of the elements are given by the
given by the coefficients but the...............power indices which also indicate the sequence
orders are given by the power................order. 1s and 0s in the coefficients signify
indices...................................................presence or absence of the elements respectively.

Examples:............................................Examples:

Natural number sequence:......................Natural number sequence:
{0,1,2,3,,,,,,,,n,,,,,,,,,,}..........................{1,1,1,1,1,,,,,,,,,1,,,,,,,,,,}

Prime number sequence:.........................Prime number sequence:
{2,3,5,7,11,13,,,,,,,,,,}...........................{0,0,1,1,0,1,0,1,0,0,0,1,0,1,,,,,,,,}
Only primes are shown...........................Both primes and composites are shown.

Applications:.......................................Applications:
Given the numeric function.....................Given the order relations
{ao, a1, a2, ,,,,, an, ,,,,,,,}.....................{0,0,1,1,0,1,0,1,0,0,0,1,0,1,,,,,,,,}
find its generating function......................find the closed form generating
and hence the recurrence.......................function. This enables more complex
relation..................................................sequences to be predicted.
=========================================================

In sequence algebra, two important properties of interest are sequence orders and duplicities of terms. If a sequence algebraist is presented with the Euler's form, he could have mistakenly interpreted F(z) as consisting of a sequence of natural numbers starting from 0 and that there are ao copies of the integer 0, a1 copies of the integer 1, a2 copies of integer 2 and so on. Obviously these are not the intended meanings in the Euler's form. To avoid confusion, readers should tune his mind to the sequence algebraic form for the rest of this paper. Note that although in sequence algebra, both power series and Laurent's seires are admitted, the latter form is favoured because of the ease of series expansion by algebraic packages and the clear visual division between duplicity factors (as numerator coefficients) and order indices of formal variables of z in the denominators. Frequently, one needs to massage the elements from the expanded series and this is conveniently done even by a manual method as shown in example 1:

Example 1: The numerator coefficient set and the denominator power index set are all displayed in separate single text lines as shown in equation (1) and these can be cut and paste for further data massaging such as collecting these into sets as shown in equation (1).

..........Nat(z):=series(1/(z-1),z=infinity,6); yields

...........................................................................These are collected into separate sets:
..........................................1......1......1......1...........1------------->set {1,1,1,1,1}
....................Nat(z):=1/z+----+----+----+----+O(----)
...........................................2.......3.......4......5...........6----------->set {1,2,3,4,5}
.........................................z.......z.......z......z...........z........................... .............(1).

Sequence algebra is based on a well known branch of applied mathematics called z-transform which is used extensively by engineers in digital signal processing and sampled data control systems. In z-transform, the strength of a unit impulse signal is represented by a Dirac's delta function, i.e., a unit impulse function [4]. A sequence of unit impulse signals with unit time interval is called a unit step impulse sequence. By changing the strength of each unit impulse and the time intervals between them, one could describe a hierachy of impulse signal sequences. However in number theoretic applications, only the model of unit impulse train is used whereby the strengths of impulses are either 1s or 0s. There is no reason why we cannot call a number sequence a signal sequence with the exception that a pure number sequence is not a time sequence. All integers are members of the natural number sequence. 1s and 0s are called duplicity factors since these indicate the number of copies of an integer. In the natural number sequence, every integer is unique and thus there is only one copy of each whether present or absent. However, when number sequences are cross multiplied or sequence summed, duplicity factors can become greater than 1 or even negative. The order of the resultant sequence is not dependent on the values of duplicity factors. To recover pure orders, these duplicity factors can be reduced to unities by using a newly defined arithmetic operator called Normc( ). In number theoretic applications, one should avoid manipulations which lead to sequence terms with negative signs or mixed signs.

It is proposed to adopt the canonical form of 1/z^i to represent the ith integer measured from the zero origin of the number sequence. Thus the integer 1 is represented by 1/z and is displaced by one unit number interval from the zero origin of 1/z^o. The numerator coefficient of unity means that there is only one copy of this term, i.e. a duplicity of 1. This representation is analogous to linear calibrations along a cartesian axis in a graph. For that reason, sequence algebra is also called visual algebra by Huen [7]. One advantage of the sequence algebraic form is that it is much simpler in finding closed forms than the Euler's form.

The starting point is the axiomatic definition that the natural number sequence can be modelled by the closed form generating function 1/(z-1). We can expand this expression either manually by long divisions or by the use of an algebraic software such as Maple V R 3 or other equivalently good symbolic softwares such as Macsymas, Reduce, and Mathematica. For consistency of notations, all algebraic sequences will be written in Maple V R 3 syntax. Equation (2) shows a finite Larent expansion of 1/(z-1) about 1/z to yield the natural number sequence. When applied to number sequences, the question of convergence seldom arise. All consecutive terms are separated by unit intervals ( not time intervals) and the power indices represent both the orders along the number sequence and the magnitudes of integers themselves. The numerator coefficients represent duplicities and for pure number sequences, these should equal unities.

Nat(z) := series(1/(z-1),z=infinity,10);

...............1......1......1......1......1.......1......1.......1.........1
......1/z+----+----+----+----+----+----+----+----+O(---)
..................2.......3.......4......5......6......7......8.......9.......10
...............z.......z.......z.........z......z......z......z.......z.........z...................(2)

A more compact way to represent the above sequence is to use a modified set notation shown in Equation (3). The fact that values of all elements are 1s means that all elements are present. Other than the natural number sequence, all other number sequences will have some integers missing so that their representations will be given by a mixture of zeroes and ones. Thus the unique representation of any number sequence can be done using this set notation as shown in equation (3).

.......Naturalnumber-set = {1,1,1,1,1,1,1,....................} ..........................(3).

The set notation in equation (3) may not be suitable for manipulations in algebraic packages but could be useful in the development of computer algorithms. Imagine the lining up of a brigade of soldiers where the places of absentees are left vacant. Such vacancies could be marked by 0s and presence marked by 1s. Further information about this absentee could be obtained once we know his order in the lineup. Thus the order property is like a pointer which points to information relating to the occupant of the ordered slot. This is the way number sequences are represented and manipulated in sequence algebra [5,6,7]. This is quite different from the Euler's form.

The ordering property above defined cannot be extended to real numbers. This is because the presence and absence of real numbers are not deterministic without further qualifications. For example, if a sequence of real numbers are presented as shown below, it is impossible to predict the missing numbers?

1.2324,3.1416,5.7674,8.6543,9.4532,10.0102 ................(4).

So even though real numbers could be viewed as a continuum between integers, the orders are indeterminate, not unless someone predefines the number of decimal digits to be used. If this predefinition is made, then the real number sequence can be converted into an integer sequence by shifting a fixed number of decimal places to the right. Following the practice in number theory, we confine discussions to natural numbers only.

Order Tests On Sequence Identities

Strictly all number sequences in sequence algebra are mixed order sequences since both odd and even number slots are included. In sequence algebra, a number sequence is never expressed with broken contiguity. Blank streches are different from zeroes as these imply that some numbers have been arbitrarily removed from the natural number sequence. This is not the way sequences are represented in sequence algebra. In conventional representations, we can write {3,5,14,21....}. In sequence algebra it should be written as {0001010000000010000001.....}. This might look like a lot of wastage of printed space but that is why we are so interested in closed form representations in sequence algebra. Fortunately, it is much easier to develop closed forms in sequence algebra than in the Euler's form.

By definition an even order sequence is one where all even terms carry unity values in the numerator coefficients and all odd terms carry zero values. The natural number sequence is a mixed order sequence where both even and odd terms carry unity values in the numerator coefficients. So is the composite number sequence. What is the order of a null sequence? Since it is neither odd nor even, it must be a mixed order sequence. If this is the interpretation, what is the order of a single zero. This does not occur in sequence algebra as the null set contains an infinity of zeroes and is thus a mixed order sequence.

Order tests is aimed at testing two types of order properties between two sequences:

(i) Order equality: Two sets {2,4,6,8,...} and (2,8,16,32...} are both of even order and have order equality but not order conformality. The second set could be a subset of the first but that is not a property relevant here.

(ii) Order conformality: The two sets {2,3,6,8,...} and {2,8,16,32,...} are not order conformal. To be order conformal, the two sets must be exactly identical.

Example 2: Check the validity of the statement assuming Even(z) to start from 2:

Order(Even(z)^2) = Order(Even(z))

The closed form expression for both are given in equations (5).

..............................Even(z) = 1/(z^2-1)

..............................Even2(z)^2 = 1/(z^2-1)^2........................................(5).

Obviously the two expressions are algebraically unequal. These can be confirmed by computer algebra as shown in equations (6) and (7):

.................................1.......1......1......1.........1
...............Even(z):=----+----+----+ ----+O(---)
...................................2......4.......6.......8.......10
................................z......z.......z.......z..........z.....................................(6).

and

....................................1.....2....3........1
.................Even2(z):=---+---+---+O(---)
......................................4....6.....8.......10
....................................z....z.....z........z................................................(7).

Even(z) and Even(z)^2 are of the same order but not order conformal.

Example 3: Order(Nat(z)^n) = Order(Nat(z)) assuming Nat(z) to start from 1.

...................................1.....1....1....1.......1
..............Nat(z) :=1/z+---+---+---+---+----.........................(8).
.....................................2.....3....4.....5......6
...................................z......z....z....z......z

and

.....................1.....2.....3....4......5.....6....5....4.....3......2....1
...Nat2(z) :=----+---+---+---+---+---+---+---+---+---+---
.......................2......3....4......5.....6....7.....8....9....10...11....12
.......................z.....z.....z.....z......z.....z....z.....z....z.......z.......z...........................(9).

Obviously Nat(z)^n = Nat(z)^(n-2) * {Nat(z)^2)} and you can proceed to complete the computations by successive squaring of Nat(z)'s. Therefore Nat(z) and Nat(z)^n are both of mixed order but are not order conformal. If however Nat(z) is defined to start from 0, all resultant sequences will be order conformal. It is important to define whether Nat(z) starts from 1 or 0 as the order properties are affected by the defintiions. Note that the numerator coefficients have increased above unities but these do not affect the order conformality property of the resultant sequence.

Example 4: Order(Goldbach(z)) = Order(Normc(Prime(z)^2))

By defintion Goldbach(z) must contain only even terms. We know that the square of odd sequence will generate only even sequences even though these may not be contiguous, i.e., there are some even terms missing. Since in Goldbach's conjecture, Prime(z) is defined to start from 3, therefore the above identity has order equality but not order conformality. Strictly, duplicity is independent of order, but Normc( ) is required if one wishes to manipulate further the generating functions of generating functions [7].

Exercise: Apply order tests to the following sequence identities assuming all sequences to start from 0.:

...............Order(Even(z) + Even(z)) = Order(Even(z))
...............Order(Even(z) + Odd(z)) = Order(Nat(z))
...............Order(Nat(z) + Prime(z)) = Order(Nat(z))
...............Order(Prime(z)+Composite(z)) = Order(Nat(z))
...............{0} * Nat(z) = {0}
...............z^(-1)*Even(z) = Odd(z)

The Composite Number Sequence

The global generating function of Composite(z) was successfully derived by Huen [5] in 1994. It is based on the sieve of Erastothenes discovered by a Greek mathematician some 2300 years ago. This ancient sieve method is designed to find primes by systematically deleting all integers divisible by integer divisors 2,3,4, .... to infinity in the natural number system. It turns out that the global composite sequence is more readily derived than the global prime sequence and to derive the former, one should adopt the reverse sieve method.

We begin the reverse sieve method by starting with a number sequence which contains only zeroes. The set or sequence is defined to start from unity which is represented by 1/z^1.

...F(z):= {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,..................}
--------------------------------------------------------------------------------------------------------------
F(z)/2:= {0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,..................}
F(z)/3:= {0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,..................}
F(z)/4:= {0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,..................}
F(z)/5:= {0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,..................}
.....................................................

We note that in general, if we choose the divisor i, each F(z)/i sequence has an initial interval shift of i intervals followed by unit impulses with uniform intervals of i-units. We say that these impulse sequences are periodic and these can be easily represented in z-transform format in closed form as follows:

........................F(z)/2 := 1/(z^2*(z^2-1));
........................F(z)/3 := 1/(z^3*(z^3-1));
........................F(z)/4 := 1/(z^4*(z^4-1));
.............................................................
........................F(z)/i := 1/(z^i*(z^i-1));..........................................(11).

We can thus easily write down the global generating function for the composite sequence as:

........................Composite(z):=1/(z^i*(z^i-1)) for i ranging from 2 to infinity.

This can be easily verified using Maple V R 3 by a single line of code as:

................series(sum(1/(z^i*(z^i-1)),i=2..upperbound),z=infinity,upperbound); ....(12).
Here is a short example written in Maple to generate 30 terms of a finite composite number sequence. Note that since composite numbers are indicated by 1s and noncomposites or primes by 0s, the actual number of terms displayed is less than 30. Note that whether one likes it or not both composite and primes are present in the output sequence. This explains the concomitant or complemnetary nature of number sequences. Using equation (6), the composite sequence is generated as shown in equation (7).

Example 5:

..................................1......2......2......1.......2.....4.....2......2.....3
...........Composite(z):=----+----+----+----+---+.---+---+---+---
....................................4........6......8......9....10...12....14....15....16
..................................z.......z......z......z.....z......z......z........z.......z

........................................4.....4......2......2.......6.....1.....2.......2.....4..........1
....................................+.---+.---+.---+.---+.---+.---+.---+ ---+.---+O(---)
.........................................18....20....21....22...24...25....26....27....28.......30
........................................z.......z.......z......z.....z......z.......z.....z......z..........z.........(13)

The numerator of each of the above term represents the number of divisors for the integer represented by the index to the order variable z in the denominator. For example 2/z^8 means that the integer 8 = 2 x 4, i.e. the two divisors are 2 and 4. On the other hand 1/z^9 shows that 9 = 3 x 3 which are nondistinct divisors and are counted as one divisor of 3. If we want to manipulate algebraically Composite(z), it is more convenient to reduce all the numerator coefficients to unities by using the arithmetic operator Normc( ). This is called the renormalising operator provided one does not confuse it with renormalisation used in theoretical physics. Equation (14) shows Normc( ) in action.

................................................1......1......1.......1.......1......1......1.......1.......1
............Normc(Composite(z)) :=---- +----+---- +---- +--- +--- +--- +--- +---
...................................................4.......6......8......9......10.....12....14....15.....16
................................................z.......z......z.......z.......z.......z......z.......z......z

......................................1.....1......1.....1.....1.....1.....1.......1......1........1
.................................+--- +--- +--- +--- +--- +--- +--- +--- +--- +O(---)
........................................18....20....21....22....24...25....26...27.....28.......30
......................................z.......z......z.......z.......z......z.....z......z......z..........z...(14).

Normc( ) is a very useful operator and is absolutely indispensible in sequence algebra. This is demonstrated by deriving the prime number sequence Prime(z) in the next section.

The Prime Number Sequence

A global generating function for the prime sequence Prime(z) is important in sequence algebra as it is the primitive sequence upon which one derives other well known sequences such as the Mersenne prime sequence, the Fermat's prime sequence, and the Twin prime sequence. For composite sequences, we have the the Perfect number sequence and the Goldbach's sequence.

Since we have already successfully derived the composite number sequence (see equation (8)), the next step is to derive the prime number sequence. This looks deceptively easy since we know it can be derived as the complementation of the composite number sequence. But this approach has its limitations since mixed mode arithmetics, which is defined here as the mixing of arithemtic and boolean logic operations, is not commutable. An alternative method which avoids the use of logical operators is to use the newly defined arithmetic operator Normc( ). This is called an arithmetic operator since its operation cannot be duplicated by using boolean logics alone.

Let us expand the close form of 1/(z-1) to 30 integer digits as shown in equation (15).

......................................1......1......1.......1......1......1......1.......1
...............Nat(z) := 1/z.+----.+----.+----+----.+---.+---.+---.+---
........................................4.......6......8.......9....10.....12....14.....15
......................................z.......z.....z.......z......z......z......z.......z

...........1......1.....1.....1......1......1......1......1......1......1..........1
.......+---.+---.+---.+---.+---.+---.+---.+---.+---.+---.+O(---)
.............16....18...20....21....22....24...25....26.....27.....28........30
...........z........z.....z.....z.......z......z.........z......z......z......z.........z

............1.......1......1......1.....1......1......1......1.......1......1
.......+----.+----.+----.+----.+---.+---.+---.+---.+---.+---
...............2......3......7......5.....11.....13....17...19....23....29
............z.......z......z.........z.....z......z.......z......z......z.......z..........................(15).

Prime(z) can be derived by using the sequence identity given in equation (16).
Prime(z):= Nat(z) - Normc(Composite(z)); .......................(16).
We obtain the prime sequence given in equation (17):
...............................1......1......1........1......1.......1......1........1
....Prime(z):=1/z.+----.+----.+----.+----.+---.+---.+---.+------
................................30.......2.......3.......7.......5......11....13......17
...............................z..........z.......z.......z.......z.......z......z.......z

...........................................1.......1.......1.........1
......................................+..---+..---+..---+O(---)
.............................................19.......23.....29.....30
............................................z........z..........z......z........................(17).

Leaving aside integers 1 which is not a prime, Prime(z) can be viewed as the complement of Composite(z).

Final Value Theorem

This is given by equation (18) where F(z) is the number sequence for which its final value is to be evaluated [4]. This theorem is useful in determining whether a number sequence is finite or infinite. Some examples are given to demonstrate its use and the pitfalls along the way.

Example 6: Find the final value of the natural number sequence Nat(z).

By defintion, Nat(z) = 1/(z-1) in sequence algebraic form.

The final value is unity! Is it true? Remember the set notation for the natural number sequence is {1,1,1,1,1,1,1,1,1,1,1,.........................}. Final value theorem gives the correct answer by stating that the numerator coefficient of 1/z^infinity is unity. This says that there is one integer with infinite value. This is quite different from the Euler's form where the final value will be expected to be infinity. Readers must be aware of this pitfall in interpretation!

Example 7: A proof that there is no end to the composite number sequence.

It is strange that Euclid was more interested in proving that there is no end to the prime number sequence than the composite number sequence. This seems to set the trend and all mathematicians after him are finding more and more alternative proofs for the prime sequence. The proof that there is no end to the composite number sequence is just as important as the proof for the prime sequence. The reason is because of the following important identities:

..............Nat(z) = Prime(z) + Normc(Composite(z)) ...................................(19).

Since there is no end to both Nat(z) and Prime(z), does it imply that the difference given by equation (20) also has no end to it?

............Normc(Composite(z)) = Nat(z) - Prime(z) ..........................................(20).

In fact the validity of equation (20) is conditional on the requirement that the difference sequence must not be prematurely truncated. For example, if Nat(z) subtracted (Nat(z) - 1/z), the resultant sequence is 1/z and is finite even though there is no end to both Nat(z) and (Nat(z)-1/z). Similarly Nat(z) - (1/z)*Nat(z)) also results in a truncated sequence. Thus it is possible that the difference between two infinite sequences could be a finite truncated sequence. The first step is to prove that the difference sequence is not truncated at the high end.

Two proofs are given where only the second proof is found to be valid.

Proof 1: We know that Prime(z) is a subset of Nat(z). We also know that no matter how large a range of Prime(z) is chosen, the minimum difference between two primes is never less than 2. Therefore it is impossible for the difference sequence to be prematurely truncated. Therefore there is no end to Normc(Composite(z)). Likewise, there is no end to Composite(z) itself. Having said this, it must be pointed out that equation (20) cannot be used to prove an infinity in Composite(z). The reason is that whereas Composite(z) can be independently derived, Prime(z) cannot be derived independent of the former [5]. Proofs using the above identity would lead to circularity and will not be valid.

Proof 2:

F(z) = composite(z) = 1/(z^i*(z^i-1))

Since 1/(z^i-1) can be factorised into 1/((z-1)*(z^(i-1)+z^(i-2)+........+z^(i-k)+....+ z + 1))). this gives

Fi(z) = 1/((z^i*(z-1)*(z^(i-1)+z^(i-2)+........+z^(i-k)+....+ z + 1)))

F2(z) = 1/(z^2*(z-1)*(z+1))
F3(z) = 1/(z^3*(z-1)*(z^2+z+1))
F4(z) = 1/(z^4*(z-1)*(z^3+z^2+z+1))
............

Therefore F(z) = sum(Fi(z),i=2..infinity);

Applying final value theorem to F(z), we get

F(z) = 1/2 + 1/3 + 1/4 + ..................+ 1/k + ........... + 1/infinity;...................(21).

This is a divergent harmonic series. Therefore the final value of F(z) is infinity. This means that there is no end to the composite series and there is an infinite number of copies of 1/z^infinity. Notice that Normc( ) has not be used. If the Normc(Composite(z)) version has been used, one will end up with one copy of 1/z^infinity which is still a valid proof but the analysis would be algebraically intractible because Normc( ) is a numerical operator.

Example 8: Find the final value of Even(z)

Even(z) = 1/(z^2-1) = 1/((z-1)*(z+1))

Final value is 1/(z*(z+1)) which has final value of 1/2. This looks strange but careful consideration will reveal that an even number sequence has alternatively numerator coefficients of 1's and 0's. Final value simply gives the average which is 1/2 as it cannot report either unity or zero.

Example 8: find the final value of Odd(z)

Odd(z) = z/(z^2-1)
Final value of Odd(z) is 1/(z+1) which is also 1/2 just as in the case of Even(z).
Check: Final value of (Even(z) + Odd(z)) = 1/2 + 1/2 = 1. This agrees with the final value of
Nat(z) = Even(z) + Odd(z).

Discrete First Order Integration

Counting the number of terms in a sequence can be done using a discrete first order integrator. Here is how this is done.

Example 10: Count the number of terms in a finite Nat(z)=sum(1/z^i,i=1..6);

.....................................1......1......1.....1......1
...........Finite(z) :=1/z+----+----+----+----+----
.........................................2.....3......4......5......6
......................................z......z.....z......z......z.................................(22)

Let first order integrator be Int1(z) = 1/(z-1) which is an infinite sequence.

After sequence cross product we get:

expand(Int1(z)*Finite(z)) =

......1......2........3.......4.......5......6.......6.......6........6........6..........6
....----+.----.+----.+----.+----.+----.+----.+----+---+-----+-------+ to infinity.
........2.......3........4........5........6......7........8.......9......10.....11.....12
......z.......z.........z........z.......z........z.......z.......z.......z.........z........z ...............(23).

The first occurrence of the largest count is at z^7. Therefore the total number of finite terms in Finite(z) is 6.

Discrete First Order Differentiation

Obviously this is the reciprocal of first order integration and is given by (z-1). If this operator is applied to equation (23), you will get back the finite sequence in equation (22). No further elaboration will be made here.

Summary

This completes a simple introduction to sequence algebra. It is fairly straightforward algebraically provided one has access to a symbolic package. One has to be awared that the defintion of generating function in sequence algebra is different from the Euler's form. The advantage of adopting this new definition is that, most of the time, one could write down closed form expressions for sequences by sight whereas in the Euler's form, the derivation could be quite elaborate. Sequence algebra represents the holistic approach to number theory which thus makes conventional number theory basically reductionistic. This means that for the first time we can handle number theory from both ends.

References

1. Brualdi A. Richard (1977): Introductory Combinatorics, Chapter 7, Generating Functions, North Holland, pp 127 to 139.

2. Liu C.L. (1985): Elements Of Discrete Mathematics, McGRAW-HILL INTERNATIONAL EDITIONS, computer science series, pp 290 to 295.

3. Charles L. Silver (1994): From Symbolic Logic ... To Mathematical Logic, WCB, Chapter 0, Mathematical Preliminaries, pp 1 to 24.

4. Luyben W.L. : Process Modeling, Simulation And Control For Chemical Engineers,1990 (second edition), International Edition, MacGraw-Hill Publishing Company, pp 402 to 404.

5. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci. Technol., 1994, VOL. 25, NO.6, pp 913-920.

6. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ. Sci. Technol., 1996, VOL.27, NO. 5, 685-691.

7. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1996, VOL.??, NO.?, ???-??? (In the press as proof paper mes 100421).

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